Transcript Dynamics Hand Out

Dynamics
FE Review Session
Adapted from the following references:
NCEES Reference Handbook v. 9.1
FE Review Manual, M.R. Lindeburg, Professional Publications
1
Kinematics
• Study of a body’s motion independent of the
forces on the body – geometry of motion.
• Fundamental kinematic equations:
2
• Rectilinear motion – motion along a straight
line
– Equations of rectilinear motion
Constant a = a0 (another place in reference)
3
• Curvilinear motion – motion along a curved
path
– Curvilinear motion coordinate systems
• Rectangular (Cartesian) coordinates
For a = constant, equations on previous slide can be
applied in the x, y, and z directions.
4
• Normal and tangential (path) coordinates
5
• Radial and transverse coordinates
6
– Special curvilinear motion situations
• Plane circular motion – motion along a circular path
Constant α = α0
7
• Projectile motion
8
• Example problems:
9
10
11
12
13
14
15
16
17
18
19
20
21
Particle Kinetics
• Kinetics is the study of motion and the forces
that cause the motion
22
• Direct application of Newton’s second law
23
• Be careful on units
– Force:
• SI: N
• US: lb
– Mass:
• SI: kg
• US: slug (lb-sec2/ft)
– Acceleration:
• SI: m/s2
• US: ft/s2
24
• Avoid the use of pound mass (lbm)
– NCEES Handbook does not appear to use lbm in
the dynamics section
– Popular study guide does use lbm
lbm
32.2 ft/sec2
25
– Rectangular coordinates

etc. for y and z
– Tangential and normal coordinates
– Radial and transverse coordinates
26
• Example problems:
27
28
29
30
31
32
33
34
35
36
37
38
• Impulse and momentum methods
– Linear impulse and momentum
t2
Gx1    Fx dt  Gx2
t1
t2
G y1    Fy dt  G y2
t1
t2
  Fdt  Linear Impulse
t1
mv  G  Linear Momentum
If
F  0
then G1  G 2
39
• Conservation of linear momentum applied to direct
central impacts
Fig 3-17 Meriam and Kraige, 7th, Wiley
40
– Angular impulse and momentum - particle
t2
HO1    M o dt  HO2
t1
Ho  r  mv Angular momentum about point O
t2
  M dt  Angular Impulse
y
o
t1
If
M
O
 0 then HO 1  HO 2
O
x
Fig 3-14 Meriam and Kraige, 7th, Wiley
41
• Work and energy methods – particle
Kinetic energy
Gravitational potential energy
Elastic potential energy
42
Example problem:
43
44
45
• Free vibrations
46
Example problem:
47
n
48
Plane Motion of a Rigid Body
• Types of rigid body motion - kinematics
Fig 5-1 Meriam and Kraige, 7th, Wiley
49
– Fixed axis rotation
v  r
2
v
an  r 
2
at  r
r
 v
v  r  ωr
a n  ω   ω  r    2r
Fig 5-1 Meriam and Kraige, 7th, Wiley
at  α  r
50
– Wheel rotating without slip
s  r
vO  r
aO  r
Fig 5-1 Meriam and Kraige, 7th, Wiley
51
– Instantaneous center of zero velocity
1. Identify directions of velocity
vectors of two points.
2. At these two points draw lines
perpendicular to the velocity
vectors.
3. These lines intersect at the
IC, point C.
Fig 5-1 Meriam and Kraige, 7th, Wiley
v A  rA
vB  rB
52
• Kinetics of plane motion of a rigid body
 F  ma
M  I
G
y
x
Fig 6/4 Meriam and Kraige, 7th, Wiley
53
Sometimes it is more
convenient to take moments
about an arbitrary point, P.
M
M
Fig 6/5 Meriam and Kraige, 7th, Wiley
P
 I   mad
P
 Iα  ρ  m a
If point P is fixed
M
P
 I P
54
FE exam format for these equations:
Parallel axis theorem:
55
– Torsional vibration:
56
57
Example problems:
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
– Work and energy applied to a rigid body
Kinetic energy
Gravitational potential energy
Elastic potential energy
73
– Impulse and momentum applied to a rigid body
74
Example problems:
75
76
77